Background: Identifying how pain transitions from acute to chronic is critical in designing effec... more Background: Identifying how pain transitions from acute to chronic is critical in designing effective prevention and management techniques for patients' well-being, physically, psychosocially, and financially. There is an increasingly pressing need for a quantitative and predictive method to evaluate how low back pain trajectories are classified and, subsequently, how we can more effectively intervene during these progression stages. Methods: In order to better understand pain mechanisms, we investigated, using computational modeling, how best to describe pain trajectories by developing a platform by which we studied the transition of acute chronic pain. Results: The present study uses a computational neuroscience-based method to conduct such trajectory research, motivated by the use of hypothalamic-pituitary-adrenal (HPA) axis activity-history over a time-period as a way to mimic pain trajectories. A numerical simulation study is presented as a "proof of concept" for this modeling approach. Conclusions: This model and its simulation results have highlighted the feasibility and the potential of developing such a broader model for patient evaluations.
Communications in Nonlinear Science and Numerical Simulation
This paper explores the internal dynamical mechanisms of epileptic seizures through quantitative ... more This paper explores the internal dynamical mechanisms of epileptic seizures through quantitative modeling based on full brain electroencephalogram (EEG) signals. Our goal is to provide seizure prediction and facilitate treatment for epileptic patients. Motivated by an earlier mathematical model with incorporated synaptic plasticity, we studied the nonlinear dynamics of inherited seizures through a differential equation model. First, driven by a set of clinical inherited electroencephalogram data recorded from a patient with diagnosed Glucose Transporter Deficiency, we developed a dynamic seizure model on a system of ordinary differential equations. The model was reduced in complexity after considering and removing redundancy of each EEG channel. Then we verified that the proposed model produces qualitatively relevant behavior which matches the basic experimental observations of inherited seizure, including synchronization index and frequency. Meanwhile, the rationality of the connectivity structure hypothesis in the modeling process was verified. Further, through varying the threshold condition and excitation strength of synaptic plasticity, we elucidated the effect of synaptic plasticity to our seizure model. Results suggest that synaptic plasticity has great effect on the duration of seizure activities, which support the plausibility of therapeutic interventions for seizure control.
ABSTRACT Using the tool of Turing instability for partial differential equations, we investigate ... more ABSTRACT Using the tool of Turing instability for partial differential equations, we investigate the spatiotemporal distributions for solutions of a predator-prey-type reaction-diffusion model with spatiotemporal delay. The linear stability conditions of Turing instability, which induce bifurcation patterns in this model, are obtained. Moreover, according to these conditions, we numerically calculate the bifurcation diagrams by using time delay and the predator rate as parameters. The effects of two parameters in the different bifurcation diagrams are also demonstrated through numerical computations and lead to some spatiotemporal patterns of this model, which enrich the pattern formation of predator-prey models.
Case reports regularly document unique or unusual aspects of glucose transporter type 1 deficienc... more Case reports regularly document unique or unusual aspects of glucose transporter type 1 deficiency (G1D). In contrast, population studies from which to draw global inferences are lacking. Twenty-five years after the earliest case reports, this deficiency still particularly affects treatment and prognostic counseling. To examine the most common features of G1D. In this study, data were collected electronically from 181 patients with G1D through a web-based, worldwide patient registry from December 1, 2013, through December 1, 2016. The study used several statistical methods tailored to address the age at onset of various forms of G1D, associated manifestations, natural history, treatment efficacy, and diagnostic procedures. These factors were correlated in a predictive mathematical model designed to guide prognosis on the basis of clinical features present at diagnosis. A total of 181 patients with G1D were included in the study (92 [50.8%] male and 89 female [49.2%]; median age, 9 y...
Height- and area-based quantitation reduce two-dimensional data to a single value. For a calibrat... more Height- and area-based quantitation reduce two-dimensional data to a single value. For a calibration set, there is a single height- or area-based quantitation equation. High-speed high-resolution data acquisition now permits rapid measurement of the width of a peak (Wh), at any height h (a fixed height, not a fixed fraction of the peak maximum) leading to any number of calibration curves. We propose a width-based quantitation (WBQ) paradigm complementing height or area based approaches. When the analyte response across the measurement range is not strictly linear, WBQ can offer superior overall performance (lower root-mean-square relative error over the entire range) compared to area- or height-based linear regression methods, rivaling weighted linear regression, provided that response is uniform near the height used for width measurement. To express concentration as an explicit function of width, chromatographic peaks are modeled as two different independent generalized Gaussian di...
Abstract In this paper, we study the existence problem of axisymmetric three‐dimensional finger s... more Abstract In this paper, we study the existence problem of axisymmetric three‐dimensional finger solutions of MullinsSekerka equation. The finger solutions are travelling wave solutions whose finger‐shaped interfaces are moving along a certain direction at a ...
A moving grid method which has its origin from differential geometry is studied. The method defor... more A moving grid method which has its origin from differential geometry is studied. The method deforms an intial grid according to a vector field calculated by a Poisson equation. The forcing term of the Poisson equation is determined by the time derivative of a positive monitor function. It adapts the grid at each time step by keeping the volume of each cell proportional to the (normalized) time-dependent monitor function. A moving finite difference method is formulated which transforms a time dependent partial differential equation by the grid mapping and then simulates the transformed equation on a fixed orthogonal grid in the computational domain. The method is demonstrated by solving model problems and an incompressible flow problem.
Andrey V. Kuzhuget, Natee Pantong and Michael V. Klibanov A globally convergent numerical method ... more Andrey V. Kuzhuget, Natee Pantong and Michael V. Klibanov A globally convergent numerical method for a Coefficient Inverse Problem with backscattering data (574K, pdf) ABSTRACT. A survey of recent results of the authors is presented. This survey is short due to space ...
Elliptic bursting arises from fast–slow systems and involves recurrent alternation between active... more Elliptic bursting arises from fast–slow systems and involves recurrent alternation between active phases of large amplitude oscillations and silent phases of small amplitude oscillations. This paper is a geometric analysis of elliptic bursting with and without noise. ...
We define a class of weakly coercive Hamiltonians and then demonstrate the single valuedness of t... more We define a class of weakly coercive Hamiltonians and then demonstrate the single valuedness of the associated Hamilton-Jacobi operators (in the viscosity sense).
In this note, we study the change of collective behavior of two synaptically coupled bursting sys... more In this note, we study the change of collective behavior of two synaptically coupled bursting systems as the strength of coupling increases. The two cells present chaotic bursting behavior when not coupled. But as the strength increases past a certain value, the behavior of two cells becomes synchronized regular bursting motions. It shows that regular oscillations can emerge from connecting intrinsically chaotic oscillators with synapses. The method of analysis is similar to that of Fast Threshold Modulation theory.
In this note, a general survey of the interesting phenomenon of delayed bifurcation is given. The... more In this note, a general survey of the interesting phenomenon of delayed bifurcation is given. These delayed bifurcations arise in particular fast-slow systems under various situations, and their mathematical justification and application are discussed. Key wordsFast-slow systems–Delay of bifurcation–Periodic forcing–Hopf bifurcation–Simple eigenvalue bifurcation–AMS(MOS) subject classifications–34C35–34D20–58F14–92C30
Background: Identifying how pain transitions from acute to chronic is critical in designing effec... more Background: Identifying how pain transitions from acute to chronic is critical in designing effective prevention and management techniques for patients' well-being, physically, psychosocially, and financially. There is an increasingly pressing need for a quantitative and predictive method to evaluate how low back pain trajectories are classified and, subsequently, how we can more effectively intervene during these progression stages. Methods: In order to better understand pain mechanisms, we investigated, using computational modeling, how best to describe pain trajectories by developing a platform by which we studied the transition of acute chronic pain. Results: The present study uses a computational neuroscience-based method to conduct such trajectory research, motivated by the use of hypothalamic-pituitary-adrenal (HPA) axis activity-history over a time-period as a way to mimic pain trajectories. A numerical simulation study is presented as a "proof of concept" for this modeling approach. Conclusions: This model and its simulation results have highlighted the feasibility and the potential of developing such a broader model for patient evaluations.
Communications in Nonlinear Science and Numerical Simulation
This paper explores the internal dynamical mechanisms of epileptic seizures through quantitative ... more This paper explores the internal dynamical mechanisms of epileptic seizures through quantitative modeling based on full brain electroencephalogram (EEG) signals. Our goal is to provide seizure prediction and facilitate treatment for epileptic patients. Motivated by an earlier mathematical model with incorporated synaptic plasticity, we studied the nonlinear dynamics of inherited seizures through a differential equation model. First, driven by a set of clinical inherited electroencephalogram data recorded from a patient with diagnosed Glucose Transporter Deficiency, we developed a dynamic seizure model on a system of ordinary differential equations. The model was reduced in complexity after considering and removing redundancy of each EEG channel. Then we verified that the proposed model produces qualitatively relevant behavior which matches the basic experimental observations of inherited seizure, including synchronization index and frequency. Meanwhile, the rationality of the connectivity structure hypothesis in the modeling process was verified. Further, through varying the threshold condition and excitation strength of synaptic plasticity, we elucidated the effect of synaptic plasticity to our seizure model. Results suggest that synaptic plasticity has great effect on the duration of seizure activities, which support the plausibility of therapeutic interventions for seizure control.
ABSTRACT Using the tool of Turing instability for partial differential equations, we investigate ... more ABSTRACT Using the tool of Turing instability for partial differential equations, we investigate the spatiotemporal distributions for solutions of a predator-prey-type reaction-diffusion model with spatiotemporal delay. The linear stability conditions of Turing instability, which induce bifurcation patterns in this model, are obtained. Moreover, according to these conditions, we numerically calculate the bifurcation diagrams by using time delay and the predator rate as parameters. The effects of two parameters in the different bifurcation diagrams are also demonstrated through numerical computations and lead to some spatiotemporal patterns of this model, which enrich the pattern formation of predator-prey models.
Case reports regularly document unique or unusual aspects of glucose transporter type 1 deficienc... more Case reports regularly document unique or unusual aspects of glucose transporter type 1 deficiency (G1D). In contrast, population studies from which to draw global inferences are lacking. Twenty-five years after the earliest case reports, this deficiency still particularly affects treatment and prognostic counseling. To examine the most common features of G1D. In this study, data were collected electronically from 181 patients with G1D through a web-based, worldwide patient registry from December 1, 2013, through December 1, 2016. The study used several statistical methods tailored to address the age at onset of various forms of G1D, associated manifestations, natural history, treatment efficacy, and diagnostic procedures. These factors were correlated in a predictive mathematical model designed to guide prognosis on the basis of clinical features present at diagnosis. A total of 181 patients with G1D were included in the study (92 [50.8%] male and 89 female [49.2%]; median age, 9 y...
Height- and area-based quantitation reduce two-dimensional data to a single value. For a calibrat... more Height- and area-based quantitation reduce two-dimensional data to a single value. For a calibration set, there is a single height- or area-based quantitation equation. High-speed high-resolution data acquisition now permits rapid measurement of the width of a peak (Wh), at any height h (a fixed height, not a fixed fraction of the peak maximum) leading to any number of calibration curves. We propose a width-based quantitation (WBQ) paradigm complementing height or area based approaches. When the analyte response across the measurement range is not strictly linear, WBQ can offer superior overall performance (lower root-mean-square relative error over the entire range) compared to area- or height-based linear regression methods, rivaling weighted linear regression, provided that response is uniform near the height used for width measurement. To express concentration as an explicit function of width, chromatographic peaks are modeled as two different independent generalized Gaussian di...
Abstract In this paper, we study the existence problem of axisymmetric three‐dimensional finger s... more Abstract In this paper, we study the existence problem of axisymmetric three‐dimensional finger solutions of MullinsSekerka equation. The finger solutions are travelling wave solutions whose finger‐shaped interfaces are moving along a certain direction at a ...
A moving grid method which has its origin from differential geometry is studied. The method defor... more A moving grid method which has its origin from differential geometry is studied. The method deforms an intial grid according to a vector field calculated by a Poisson equation. The forcing term of the Poisson equation is determined by the time derivative of a positive monitor function. It adapts the grid at each time step by keeping the volume of each cell proportional to the (normalized) time-dependent monitor function. A moving finite difference method is formulated which transforms a time dependent partial differential equation by the grid mapping and then simulates the transformed equation on a fixed orthogonal grid in the computational domain. The method is demonstrated by solving model problems and an incompressible flow problem.
Andrey V. Kuzhuget, Natee Pantong and Michael V. Klibanov A globally convergent numerical method ... more Andrey V. Kuzhuget, Natee Pantong and Michael V. Klibanov A globally convergent numerical method for a Coefficient Inverse Problem with backscattering data (574K, pdf) ABSTRACT. A survey of recent results of the authors is presented. This survey is short due to space ...
Elliptic bursting arises from fast–slow systems and involves recurrent alternation between active... more Elliptic bursting arises from fast–slow systems and involves recurrent alternation between active phases of large amplitude oscillations and silent phases of small amplitude oscillations. This paper is a geometric analysis of elliptic bursting with and without noise. ...
We define a class of weakly coercive Hamiltonians and then demonstrate the single valuedness of t... more We define a class of weakly coercive Hamiltonians and then demonstrate the single valuedness of the associated Hamilton-Jacobi operators (in the viscosity sense).
In this note, we study the change of collective behavior of two synaptically coupled bursting sys... more In this note, we study the change of collective behavior of two synaptically coupled bursting systems as the strength of coupling increases. The two cells present chaotic bursting behavior when not coupled. But as the strength increases past a certain value, the behavior of two cells becomes synchronized regular bursting motions. It shows that regular oscillations can emerge from connecting intrinsically chaotic oscillators with synapses. The method of analysis is similar to that of Fast Threshold Modulation theory.
In this note, a general survey of the interesting phenomenon of delayed bifurcation is given. The... more In this note, a general survey of the interesting phenomenon of delayed bifurcation is given. These delayed bifurcations arise in particular fast-slow systems under various situations, and their mathematical justification and application are discussed. Key wordsFast-slow systems–Delay of bifurcation–Periodic forcing–Hopf bifurcation–Simple eigenvalue bifurcation–AMS(MOS) subject classifications–34C35–34D20–58F14–92C30
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